A numerical approach for solving a class of fractional optimal control problems via operational matrix Bernoulli polynomials

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Abstract
The purpose of this study is to introduce a novel approach based on the operational matrix of a Riemann–Liouville fractional integral of Bernoulli polynomials, in order to numerically solve a class of fractional optimal control problems that arise in engineering. The method is computationally consistent and moreover, it has good flexibility in satisfying the initial and boundary conditions. The fractional derivative in the dynamic system is considered in the Caputo sense. The upper bound of the error for function approximation by a Bernoulli polynomial is also given. In order to numerically solve the given problem, the problem is transformed into a functional integral equation that is equivalent to the given problem. Then, the new integral equation is approximated by utilizing the Gauss quadrature formula. Afterwards, a system of nonlinear equations is yielded from the Lagrange multipliers method. Finally, the resultant system of nonlinear equations is solved by Newton’s iterative method. Some illustrative examples are included to demonstrate the applicability of the new technique.

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